# Objective Set Compression

Test-Based Problems and Multiobjective Optimization
• Edwin D. de Jong
• Anthony Bucci
Chapter
Part of the Natural Computing Series book series (NCS)

## Summary

We consider a class of optimization problems wherein the quality of candidate solutions is estimated by their performance on a number of tests. Classifier induction, function regression, and certain types of reinforcement learning, including problems often attacked with coevolutionary algorithms, can all be seen as members of this class. Traditional approaches to such test-based problems use a single objective function that aggregates the scores obtained on the tests. Recent work, by contrast, argues that useful finer-grained distinctions between candidate solutions are obtained when each test is treated as a separate objective, and that algorithms employing such multiobjective comparisons show favourable behaviour relative to those which do not. Unfortunately, the number of tests can be very large. Since it is well-known that high-dimensional multiobjective optimization problems are difficult to handle in practice, the question arises whether the multiobjective treatment of test-based problems is feasible. To begin addressing this question, we examine a method for reducing the number of dimensions without sacrificing the favorable properties of the multiobjective approach. Our method, which is a form of dimension extraction, finds underlying objectives implicit in test-based problems. Essentially, the method proceeds by placing the tests along the minimal number of coordinate axes that still preserve ordering information among the candidate solutions. Application of the method to the strategy set for several instances of the game of Nim suggest the technique has significant practical benefits: a type of compression of the set of objectives is observed in all tested instances. Surprisingly, we also find that the information contained in the arrangement of tests on the coordinate axes reveals important information about the structure of the underlying problem.

## Keywords

Coordinate System Multiobjective Optimization Candidate Solution Multiobjective Optimization Problem Initial Objective

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